Atonal music analysis by proportional graph
This paper shows how do pitch and rhythmic structures interacts in the Webern atonal music to produce a coherent and unified work. This is an interesting and difficult problem to approach. Because of there is no Webern's characteristic routine at the surface level of his music. However it will be shown that beneath the complex surface exists a considerable degree of regularity. This paper shows the proportional graph. Its purpose is to present rhythmic structure in the clearest possible way, detached from ordinary notation with its bias toward traditional interpretation. From the proportional graph and A-R partition, we see the correspondences and relations of various types demonstrated in the atonal composition of Webern. Using the composite segmentation, the pitch class set structure of the atonal work is related to the fundamental rhythmic structures of the work. The composite segmentation represents the harmonic components of the music at the absolute surface level and the segmentation which may take into account than the single voice, which does not encompass the total texture from top th bottom for any segment, and which may involve overlappings. The pitch-class set inclusion relations match the pattern of proportional relations set out in the basic rhythmic structure pattern. The proportion is quite explicitly reflected by the partition. The relations among pitch-class sets in terms of intersections, unions, complements have structural analogues in the relations among rhythmic formations in terms of combinations and partitions of duration. Proportional graph and A-R partition are the tool of atonal music and rhythm structure analysis. we can see the relation between pitch and rhythm at the surface level. However we can look forward to see the further study of pitch and rhythm relation at the middle ground or background level.